summation - Simple proof by induction: $1^3 + 2^3 +3^3 +...+ n^3 = [ (n(n+1))/2 ]^2$

I am rather illiterate when it comes to mathematics, I am afraid. In an effort to change that, I grabbed a copy of 'What is mathematics? : An elementary approach to ideas and methods' and have already encountered some difficulty. It seems silly to waste so much time trying to solve it myself, so I decided to ask for some help in solving and illuminating the taken steps, so that I can solve some more on my own. I shall review some of the basics while awaiting answers, and hope that my session tomorrow shall be more productive...

The problem is as follows:

Prove by induction that $1^3 + 2^3 +3^3 ... n^3 = [ (n(n+1))/2 ]^2$.

As is, at the point I decided to seek help and look up material for review, I have taken the following steps:

solved for the base case: n=1, $1^3 = [2/2]^2$

$$1 = 1^2, 1=1$$

Then after proving the basis, I stated the assumption that:

$1^3 + 2^3 ... k^3 = [ (k(k+1)) / 2 ]^2$ is true.

Then I tried to solve for the next case, $(k+1)^3$:

$$1^3 + 2^3 + 3^3 ... k^3 + (k+1)^3 = [ ( (k(k+1)) / 2 ) + (k+1) ]^2$$

$$[ (k(k+1)) / 2 ]^2 + (k+1)^3 = [ (k(k+1) + 1(k+1) ) / 2 ]^2$$

at this point after failing for a while and having spent quite some time looking up material, I sought help. I hope I'm going down the right path here, but I suspect I shall soon find out...

Help is greatly appreciated.